Problem 27 - Fundamentals of Physics

Question

A particle moves along the \(x^{\prime}\) axis of frame \(S^{\prime}\) with velocity \(0.40 c\). Frame \(S^{\prime}\) moves with velocity \(0.60 c\) with respect to frame \(S\). What is the velocity of the particle with respect to frame \(S ?\)

Step-by-Step Solution

Answer

The particle's velocity with respect to frame \(S\) is approximately \(0.806c\).

Step 1: Understand the Problem

We have a particle moving with velocity 0.40c in a frame \(S^{\prime}\). Frame \(S^{\prime}\) itself is moving with velocity 0.60c relative to another frame \(S\). We need to find the velocity of the particle relative to frame \(S\).

Step 2: Apply the Lorentz Velocity Transformation Formula

To find the velocity \(v\) of the particle in frame \(S\), we use the relativistic velocity addition formula: \[v = \frac{v^{\prime} + u}{1 + \frac{v^{\prime} u}{c^2}}\] where \(v^{\prime} = 0.40c\) is the velocity of the particle in \(S^{\prime}\), and \(u = 0.60c\) is the velocity of \(S^{\prime}\) with respect to \(S\).

Step 3: Plug Values Into the Formula

Substitute \(v^{\prime} = 0.40c\) and \(u = 0.60c\) into the formula: \[v = \frac{0.40c + 0.60c}{1 + \frac{0.40c \times 0.60c}{c^2}}\]

Step 4: Simplify the Numerator

Calculate the numerator: \(0.40c + 0.60c = 1.00c\).

Step 5: Simplify the Denominator

Calculate the denominator: \[1 + \frac{0.40c \times 0.60c}{c^2} = 1 + 0.24 = 1.24\]

Step 6: Calculate the Final Result

Now, compute \(v\): \[v = \frac{1.00c}{1.24} = 0.806c\]. This is the velocity of the particle relative to frame \(S\).

Key Concepts

Relativistic Velocity Addition

In the realm of special relativity, traditional rules of adding velocities do not directly apply when dealing with speeds that are significant fractions of the speed of light. Instead, we use the concept of relativistic velocity addition. This formula helps us calculate how velocities add up when observing from different frames of reference, particularly when relativistic effects become significant.

For two velocities \( v' \) and \( u \), the formula is given by:

• \( v = \frac{v' + u}{1 + \frac{v'u}{c^2}} \)

Here, \( v' \) is the velocity of, for example, a particle in a moving frame, \( u \) is the velocity of the frame itself with respect to another frame, and \( c \) is the speed of light. The denominator accounts for the relativistic effect that ensures no object will ever exceed the speed of light.

This approach corrects our intuitive but non-relativistic velocity addition to ensure compliance with the principles of special relativity.

Frames of Reference

In physics, a frame of reference is a set of coordinates or a viewpoint used to measure and observe physical phenomena. Different observers may have different frames of reference, which can change how motion and velocity are perceived. Depending on the observer's state of motion—whether at rest or moving—observations about the speed or position of an object can vary.

In our example exercise, frame \( S' \) is moving relative to frame \( S \), which affects how velocities are observed from each frame. The relativistic velocity addition is necessary because simple addition of velocities from classical mechanics doesn't hold in these scenarios.

The concept of frames of reference is pivotal in special relativity as it emphasizes that the laws of physics are the same in all inertial frames, and how these frames interact is crucial in accurately describing motion.

Special Relativity

Special relativity is a theory proposed by Albert Einstein, which revolutionized our understanding of space, time, and motion. It asserts that the laws of physics are invariant in all inertial frames, and it introduces the concept that the speed of light is the universe's ultimate speed limit.

Some key ideas in special relativity include:

• Time dilation: Time is not absolute but can differ for observers in different frames of reference.
• Length contraction: Objects may appear shorter in the direction of motion from the perspective of moving observers.

In the exercise provided, special relativity governs the approach we take by utilizing the relativistic velocity addition formula. This ensures that calculated velocities respect the cosmic speed limit, maintaining consistency with both the actual observed speeds and the theoretical framework established by Einstein.

Understanding these principles is fundamental in modern physics, especially in high-speed and high-energy applications where relativistic effects are non-negligible.